{
  "id": "2002.11123",
  "version": "v1",
  "published": "2020-02-25T19:00:01.000Z",
  "updated": "2020-02-25T19:00:01.000Z",
  "title": "Diffusive shock acceleration in $N$ dimensions",
  "authors": [
    "Assaf Lavi",
    "Ofir Arad",
    "Yotam Nagar",
    "Uri Keshet"
  ],
  "comment": "14 pages, 7 figures, comments welcome",
  "categories": [
    "astro-ph.HE"
  ],
  "abstract": "Collisionless shocks are often studied in two spatial dimensions (2D), to gain insights into the 3D case. We analyze diffusive shock acceleration for an arbitrary number $N\\in\\mathbb{N}$ of dimensions. For a non-relativistic shock of compression ratio $\\mathcal{R}$, the spectral index of the accelerated particles is $s_E=1+N/(\\mathcal{R}-1)$; this curiously yields, for any $N$, the familiar $s_E=2$ (i.e., equal energy per logarithmic particle energy bin) for a strong shock in a mono-atomic gas. A precise relation between $s_E$ and the anisotropy along an arbitrary relativistic shock is derived, and is used to obtain an analytic expression for $s_E$ in the case of isotropic angular diffusion, affirming an analogous result in 3D. In particular, this approach yields $s_E = (1+\\sqrt{13})/2 \\simeq 2.30$ in the ultra-relativistic shock limit for $N=2$, and $s_E(N\\to\\infty)=2$ for any strong shock. The angular eigenfunctions of the isotropic-diffusion transport equation reduce in 2D to elliptic cosine functions, providing a rigorous solution to the problem; the first function upstream already yields a remarkably accurate approximation. We show how these and additional results can be used to promote the study of shocks in 3D.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-25T19:00:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dimensions",
      "isotropic-diffusion transport equation reduce",
      "logarithmic particle energy bin",
      "strong shock",
      "analyze diffusive shock acceleration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}